Problem Bank

In $\triangle ABC$, $AB=2AC$ and $\angle BAC=60^\circ$. An interior point $P$ satisfies $PA=\sqrt 3$, $PC=2$, $PB=5$. Find the area of $\triangle ABC$.

Evaluate

The function

is odd, with domain all of $\mathbb{R}$. Find $a$ and $b$.

The ellipse $C_1: \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ $(a > b > 0)$ has eccentricity $\dfrac{\sqrt{2}}{2}$, and …

As shown in the figure, in rectangle $ABCD$ the points $E$ and $F$ lie on sides $AB$ and $BC$ respectively. Triangle $DEF$ …

Town $B$ is $4$ km due east of town $A$. Town $C$ is $2$ km from $B$ in the direction N$30^\circ$E. Along the riverbank, mode…

Let $f(x) = a\mathrm{e}^x$ and $g(x) = \ln x + b$ with $a, b \in \mathbb{R}$.

1. When $b = 1$, if $f(x) \geqslant g(x)$ holds for all $x > 0$, find the range of possible values of…

Complex numbers $z, w$ satisfy $z + w = 1$ and $zw = \mathrm{i}$, where $\mathrm{i}$ is the imaginary unit. Find $z^5 + w^5$.

A sequence $\{a_n\}$ has $a_1 = 1$, and for $n \geqslant 2$,

Let $f(x) = \mathrm{e}^x + \dfrac{1}{2}x^2 + ax$.

1. If $a = -1$, find the intervals on which $f(x)$ is monotonic. 2. If $a \in [0, 1]$, prove that

(You may us…

Two walls meet at a right angle. A fence of total length $60$ meters, consisting of two straight segments, is placed so that the two fence segments and the two walls bound a quadri…

For $x, y \in [0, 1]$, find the range of